Five-Dimensional Gauge Theories and Local Mirror Symmetry

We study the dynamics of 5-dimensional gauge theory on $M_4\times S^1$ by compactifying type II/M theory on degenerate Calabi-Yau manifolds. We use the local mirror symmetry and shall show that the prepotential of the 5-dimensional SU(2) gauge theory without matter is given exactly by that of the type II string theory compactified on the local ${\bf F}_2$, i.e. Hirzebruch surface ${\bf F}_2$ lying inside a non-compact Calabi-Yau manifold. It is shown that our result reproduces the Seiberg-Witten theory at the 4-dimensional limit $R\to 0$ ($R$ denotes the radius of $S^1$) and also the result of the uncompactified 5-dimensional theory at $R\to \infty$. We also discuss SU(2) gauge theory with $1\le N_f\le 4$ matter in vector representations and show that they are described by the geometry of the local ${\bf F}_2$ blown up at $N_f$ points.